Publication: Towards a theory of chaos explained as travel on Riemann surfaces
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2009-01-09
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IOP Publishing Ltd
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Abstract
We investigate the dynamics defined by the following set of three coupled first-order ODEs: (z) over dot (n) + i omega z(n) = g(n+2)/z(n) - z(n+1) + g(n+1)/z(n) - z(n+2) It is shown that the system can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable), and we investigate the position and nature of its branch points. In the semi-symmetric case (g(1) = g(2) not equal g(3)), for rational values of the coupling constants the system is isochronous and explicit formulae for the period of the solutions can be given. For irrational values, the motions are confined but feature aperiodic motion with sensitive dependence on initial conditions. The system shows a rich dynamical behaviour that can be understood in quantitative detail since a global description of the Riemann surface associated with the solutions can be achieved. The details of the description of the Riemann surface are postponed to a forthcoming publication. This toy model is meant to provide a paradigmatic first step towards understanding a certain novel kind of chaotic behaviour.
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© IOP Publishing Ltd.
We would like to thank the Centro Internacional de Ciencias in Cuernavaca, in particular François Leyvraz and Thomas Seligman, for their support in organizing the Scientific Gatherings on Integrable Systems and the Transition to Chaos which provided several opportunities for us to meet and work together. It is a pleasure to acknowledge illuminating discussions with Boris Dubrovin, Yuri Fedorov, Jean-Pierre Françoise, Peter Grinevich, François Leyvraz, Alexander Mikhailov, Thomas Seligman and Carles Simó.
The research of DGU is supported in part by the Ramón y Cajal program of the Spanish ministry of Science and Technology and by the DGI under grants MTM2006- 00478 and MTM2006- 14603.
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